Deep Learning of Compositional Targets with Hierarchical Spectral Methods

Hugo Tabanelli; Yatin Dandi; Luca Pesce; Florent Krzakala

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arXiv · 2602.10867

Deep Learning of Compositional Targets with Hierarchical Spectral Methods

Hugo Tabanelli·Yatin Dandi·Luca Pesce·Florent Krzakala

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Abstract

Why depth yields a genuine computational advantage over shallow methods remains a central open question in learning theory. We study this question in a controlled high-dimensional Gaussian setting, focusing on compositional target functions. We analyze their learnability using an explicit three-layer fitting model trained via layer-wise spectral estimators. Although the target is globally a high-degree polynomial, its compositional structure allows learning to proceed in stages: an intermediate representation reveals structure that is inaccessible at the input level. This reduces learning to simpler spectral estimation problems, well studied in the context of multi-index models, whereas any shallow estimator must resolve all components simultaneously. Our analysis relies on Gaussian universality, leading to sharp separations in sample complexity between two and three-layer learning strategies.

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Hugo Tabanelli, Yatin Dandi, Luca Pesce, Florent Krzakala. 2026-02-11. Deep Learning of Compositional Targets with Hierarchical Spectral Methods. https://arxiv.org/abs/2602.10867

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